The statistical non-ergodicity of the eigenstates of the quantum chaotic systems and its semi-classical limit

In the semi-classical limit, the non-ergodicity of the eigenstates, theta(k)(j), of circular unitary ensemble (CUE) are investigated. To study statistically the non-ergodicity of the eigenstates, of a quantum system, a pair of statistical functions, Phi(N)(j) = Sigma(k=0)(N-1)\theta(k)(j)\(4) and Psi(N)(j) = Sigma(k=0)(N-1)root\theta k(j)\(2), are defined to show the scars and anti-scars respectively. In the frame of random Matrix Theory, Phi(N)(j)s and Psi(N)(j)s for random orthohormal unit vectors are calculated. It is shown that their averages and fluctuations will tend to zero with the increase of N, while they follow the scaling laws. Compared with Phi(N)(j)s and Psi(N)(j)s Obtained from the eigenstates of the quantum baker's transformation, it is found that, with the presence of scars (or antiscars), the fluctuations of the statistical functions of the eigenstates of the quantum baker's transformation will be greater than those of the random matrices, and tend to zero much slower in the semi-classical limit of N --> infinity